Natural Science Forum / Physics / Research / January 2008

Forget my previous reply; it was just plain silly. I made a (huge)
mistake about the meaning of "non-degenerate". The quadratic form q
on C^n defined by

   q(z_1,...z_n) = sum_k(z_k)2

*is* non-degenerate.

The problem lies elsewhere. Varadarajan defines Spin(n,C) (on page 193)
as the universal cover of SO(n,C). Then he proves (on page 198) that
Spin(n,C) is isomorphic to the group of the invertible elements _x_ of
the Clifford algebra Cl(C^n,q) such that

   x.C^n.x^{-1} is a subset of C^n   and   x.beta(x) = 1,

where beta is the principal antiautomorphism of Cl(C^n,q).

On the other hand,  Lawson and Michelsohn define Spin(n,C) (on page 18)
as the group of invertible elements Cl(C^n,q) generated by those
elements of the form

   v_1 v_2 ... v_n

where each v_k belongs to C^n, _n_ is even and q(v_k) = 1 or -1 for each
_k_.

Therefore, they are not the same groups! If _v_ is an element of C^n
such that q(v) = -1, it belongs to the group that Lawson and Michelsohn
are talking about, but not the group that Varadarajan is talking about.
(However, they would be the same group over the reals; in that case, the
possibility q(v) = -1 does not occur.) The group considered by Lawson
and Michelson has, so to speak, twice as many elements as the one
considered by Varadarajan.

Best regards,

Jose Carlos Santos